Works › Prior Analytics, Books I–II
Volume 1 · pp. 713–724
Book II of the Prior Analytics, End of Tractate I and Tractate II, Chapters I-IV (part)
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(quod est esse magnum) sit ex necessitate, sequitur enim si album est, magnum est, eo quod magnum est, subjectum albi cadens in diffinitione ejus: album enim non est nisi in magnitudine, sicut simum non est nisi in naso: unde sicut sequitur, si simum est, nasus est: ita sequitur, si A album est, B magnum necesse est esse: et cum hoc quod dicatur, A album non est: ad haec enim duo A esse et non esse, impossibile est idem consequens sequi, scilicet quod sit B magnum. Quando enim ponitur inter tres terminos, consequentia scilicet, quod quando est A album positum, necesse est B magnum esse: et cum est B magnum, sequatur necessario C non album esse: sequitur necessario a primo ad ultimum, quod si A album est, quod etiam C album non est: quod tamen est impossibile: sed sumitur ex adversario concedente, quod idem consequens sequitur ad antecedens esse et non esse: et tunc quidem a primo ad ultimum est in duobus sicut in tribus.
Quando igitur duobus existentibus (antecedente scilicet et consequente) cum unum eorum (scilicet antecedens) sit, necesse est alterum esse, quod est consequens: et si hoc (scilicet consequens) non est, necesse alterum non esse quod est antecedens: cum ergo B magnum (quod est consequens) non sit A album (quod est antecedens) non potest poni esse: quia destructo consequente destruitur antecedens. Si ergo dicatur, quod cum A album non sit, necesse sit B magnum esse (quod est consequens: et hoc dicit qui concedit, quod ad destructionem antecedentis necessario sequitur consequens) sequitur et accidit de necessitate a primo ad ultimum arguendo, quod cum B magnum non sit, idem B magnum sit 1: si enim idem est consequens ad esse et non esse antecedentis, sequitur autem si antecedens est, consequens est: et si consequens non est, antecedens non est: sequitur ad idem esse, esse consequens et ad non esse, esse consequens: ergo a primo ad ultimum a non esse consequens sequitur esse consequens: hoc autem est impossibile: nam in consequentiis talibus destructo consequente destruitur antecedens: unde si B magnum non est, A album antecedens non est, ex necessitate consequentiae. Si igitur detur quod cum non sit antecedens, adhuc erit consequens, sicut si dicam cum non sit A album antecedens, adhuc erit B magnum quod est consequens, accidit per duo sicut per tria, quod si B magnum non est, quod B magnum est: esse enim accidit ex positione antecedentis, non esse autem ex hypothesi concedentis, quod idem est consequens sequens ad positionem antecedentis et destructionem antecedentis.
Et ex hoc concluditur quod verum ex falso non sequitur per syllogismum dicentem propter quid ut causam conclusionis, sed per syllogismum quia: et sic ad idem esse et non esse, non sequitur aliquid ex necessitate esse. Sed ad antecedens esse vel esse verum sequitur consequens esse vel esse verum, quod idem est: ergo ad antecedens non esse vel esse falsum, quod idem est, non sequitur consequens esse vel esse verum, quod idem est. Et sic non ex necessitate causae dicente propter quid sequitur verum ex falso: quia si consequens non est, antecedens non est: et si idem consequens sequitur ad esse et non esse antecedentis: si antecedens non est, consequens est: ergo a primo ad ultimum: si consequens non est, consequens est: quod est impossibile: et sic probatum est propositum.
TRACTATUS II.
DE SYLLOGISMO CIRCULARI.
CAPUT I.
Cujus sit potestatis, et an syllogismus, et ad quem usum, et quid sit per diffinitionem syllogismus circularis.
Post haec de syllogismo circulari tractandum est: quia in illo potestas syllogizandi consideratur, quae est ex conclusione super praemissas. In praecedentibus potestas syllogizandi considerata est, quae est ex praemissis super conclusiones duas, vel unam, vel secundum quod aliquid potest inferri ex praemissis. Primum attendendum est, quod quamvis non sit de arte vel natura, quod idem prius et posterius sit seipso, et aliquid notius et ignotius seipso: tamen idem sub uno modo acceptum et sub alio modo acceptum potest esse notius et ignotius seipso quantum ad nos, et prius et posterius, sicut est videre in diffinitione et diffinito, quorum utrumque infert reliquum, cum tamen idem sint: et sic est in conversa et convertente, et in multis aliis: et hoc modo fit in circulari syllogismo, sicut patebit in sequentibus.
which is to be great, is by necessity. For it follows, if it is white, that it is great, because great is the subject of white, falling in its definition: for white exists only in magnitude, just as snub exists only in a nose. Hence, just as it follows, if the snub exists, a nose exists, so it follows, if A is white, that B must be great. And when along with this it is said that A is not white, then to these two, namely A's being and not being, it is impossible for the same consequent to follow, namely that B is great. For when there is posited among three terms the consequence, namely that when A white is posited, B great must be, and when B is great, C non-white follows necessarily, it follows necessarily from first to last that, if A is white, then also C is not white; and this is impossible. But it is taken from the adversary granting that the same consequent follows upon the antecedent's being and not being; and then, from first to last, the case is in two things as in three.
Therefore, when two things exist, namely antecedent and consequent, when one of them, namely the antecedent, exists, the other must exist, which is the consequent; and if this, namely the consequent, does not exist, the other, which is the antecedent, must not exist. Since, therefore, B great, which is the consequent, does not exist, A white, which is the antecedent, cannot be posited to exist, because when the consequent is destroyed the antecedent is destroyed. If, therefore, it is said that, when A white does not exist, B great must exist, which is the consequent, and this is what one says who grants that upon the destruction of the antecedent the consequent necessarily follows, then it follows and occurs of necessity, by arguing from first to last, that when B great does not exist, that same B great exists 1. For if the same thing is consequent upon the being and non-being of the antecedent, but it follows that if the antecedent exists the consequent exists, and if the consequent does not exist the antecedent does not exist, then upon the same being there follows the being of the consequent, and upon non-being there follows the being of the consequent; therefore, from first to last, from the non-being of the consequent there follows the being of the consequent. But this is impossible; for in such consequences, when the consequent is destroyed the antecedent is destroyed. Hence if B great does not exist, A white, the antecedent, does not exist, by the necessity of the consequence. Therefore, if it is granted that when the antecedent does not exist the consequent will still exist, as if I say that when A white, the antecedent, does not exist, B great, which is the consequent, will still exist, it happens through two as through three that, if B great does not exist, B great exists. For being occurs from the positing of the antecedent, but non-being from the hypothesis of the one granting that the same thing is consequent following upon the positing of the antecedent and the destruction of the antecedent.
And from this it is concluded that truth from falsehood does not follow through a syllogism stating propter quid, as the cause of the conclusion, but through a syllogism quia. Thus upon the same being and non-being, something's existing by necessity does not follow. But upon the antecedent's being, or being true, there follows the consequent's being, or being true, which is the same thing; therefore upon the antecedent's not being, or being false, which is the same thing, there does not follow the consequent's being, or being true, which is the same thing. And thus truth does not follow from falsehood from the necessity of a cause stating propter quid, because if the consequent does not exist, the antecedent does not exist; and if the same consequent follows upon the being and non-being of the antecedent, if the antecedent does not exist, the consequent exists; therefore, from first to last: if the consequent does not exist, the consequent exists, which is impossible. And thus the proposition has been proved.
TRACTATE II.
ON THE CIRCULAR SYLLOGISM.
CHAPTER I.
Of what power it is, and whether it is a syllogism, and to what use it is ordered, and what a circular syllogism is by definition.
After these things one must treat of the circular syllogism, because in it the power of syllogizing is considered, which is from the conclusion back upon the premises. In what preceded, the power of syllogizing was considered which is from the premises upon two conclusions, or one, or according as something can be inferred from the premises. First it must be attended to that, although it is not according to art or nature that the same thing should be prior and posterior to itself, and that something should be better known and less known than itself, nevertheless the same thing, taken under one mode and taken under another mode, can be better known and less known than itself with respect to us, and prior and posterior, as one can see in definition and the defined, each of which infers the other, although they are the same. And so it is in the converted proposition and the converting one, and in many other cases; and in this way it occurs in the circular syllogism, as will be clear in what follows.
Adhuc autem si quis objiciat quod in omni syllogismo est decursus ab universali ad particulare: in circulo autem de pari in par decurritur. Facile est respondere: quia sicut propositio constituitur ex convertibilibus, et tamen subjectum est in ratione partis, et praedicatum in ratione totius: ita fit decursus in syllogismo in terminis convertibilibus, et tamen major est in ratione universalis, et conclusio in ratione partis praemissarum: et non oportet in syllogismo secundum quod syllogismus est, talem decursum esse secundum rem, sed secundum modum tantum, secundum signa distributiva acceptum 2. Hic autem syllogismus circularis utilis est et demonstratori, et dialectico: demonstrator enim ostendit per causam immediatam effectum essentialem, et e converso causam per talem effectum: et sic reflectetur circulus si a primo ad ultimum praedicatur: si enim per A causam ostenditur B effectus, et per B effectum ostenditur A causa: ergo a primo ad ultimum A causa ostenditur per seipsam. Dialectico autem est utilis: quia ille procedit ex opinatis: et sufficit ei idem per idem ostendere secundum rem, dummodo respondenti vel aliis videatur notius esse sub uno modo quam sub alio.
Si quis autem objiciat quod in tali processu videtur esse petitio ejus quod est in principio, dicendum quod non est: petitio enim principii non est nisi in propositione, quae non est nata per se ostendi, et ostenditur per seipsam vel per aeque dubiam sibi. In istis autem propositionibus quae sunt natae per seipsas ostendi, non fit petitio ejus quod est in principio; et tales sunt propositiones quae sunt in terminis convertibilibus, quae licet secundum rem idem sint, non tamen secundum modum accipiuntur ut idem, sed ut notius uno modo, et ignotius alio modo.
Tractabimus ergo hic de syllogismo circulari secundum quod in communi consideratus abstrahit a terminis causalibus et opinabilibus: quia talis consideratio pertinet ad hanc scientiam. In secundo autem Posteriorum secundum quod in terminis causalibus consideratur, agendum erit de ipso.
Dicamus igitur quod idem est circulo sive circulariter aliquid ostendere, ei quod est ex se invicem ostendere: et hoc quidem (per diffinitionem) est per conclusionem facti syllogismi et praedicationem unam sive praemissam propositionem acceptam in situ terminorum e converso (sicut fit in omni conversa in terminis propositione) concludere reliquam sive majorem sive minorem, cujus conversa cum conclusione non ponitur: et haec est reliqua praemissarum, quam prius sumpserat in primo syllogismo non reflexo. Cujus exemplum est: si enim velimus ostendere, quomodo ex conclusione et conversa minoris syllogizatur major, fiat sic principalis et directe procedens syllogismus: omne B est A, omne C B, ergo omne C A. Deinde sumatur ad reflexum syllogismum illa major quae fuit conclusio, scilicet omne C est A, cum conversa minoris quae est omne B C, et inferatur conclusio quae fuit major prioris syllogismi, sic, omne C A, omne B C, ergo omne B A. Unde si oportet monstrare majorem, hanc scilicet, quoniam A inest omni C, ostendat hoc aliquis per B medium quod in priori syllogismo fuit minus extremum.
Again, if someone objects that in every syllogism there is a descent from the universal to the particular, but in a circle the descent is from equal to equal, it is easy to answer: because just as a proposition is constituted from convertible terms, and nevertheless the subject is in the account of a part and the predicate in the account of the whole, so the descent in a syllogism occurs in convertible terms, and nevertheless the major is in the account of the universal and the conclusion in the account of part of the premises. And in a syllogism insofar as it is a syllogism, such descent need not exist according to reality, but only according to mode, taken according to distributive signs 2. This circular syllogism is useful both to the demonstrator and to the dialectician. For the demonstrator shows the essential effect through the immediate cause, and conversely the cause through such an effect; and thus the circle will be reflected if it is predicated from first to last. For if through A as cause B as effect is shown, and through B as effect A as cause is shown, then from first to last A as cause is shown through itself. But it is useful to the dialectician because he proceeds from things opined; and for him it is enough to show the same through the same according to reality, provided that to the respondent or to others it seems better known under one mode than under another.
But if someone objects that in such a process there seems to be a begging of that which is at the beginning, one must say that there is not. For begging the principle occurs only in a proposition which is not naturally fitted to be shown through itself, and is shown through itself or through something equally doubtful to it. But in those propositions which are naturally fitted to be shown through themselves, there is no begging of what is in the beginning; and such are propositions which are in convertible terms, which, although they are the same according to reality, nevertheless are not taken as the same according to mode, but as better known in one mode and less known in another mode.
Therefore we shall treat here of the circular syllogism insofar as, considered in common, it abstracts from causal and opinable terms, because such consideration belongs to this science. But in the second book of the Posterior Analytics, insofar as it is considered in causal terms, it will have to be treated.
Therefore let us say that to show something in a circle, or circularly, is the same as to show from things mutually through themselves; and this, by definition, is to conclude the remaining premise, whether the major or the minor, through the conclusion of the syllogism that has been made and one predication or premise-proposition accepted in the converse position of terms, as happens in every proposition converted in its terms, whose converse is not posited with the conclusion. And this is the remaining one of the premises which previously had been taken in the first non-reflexive syllogism. An example is this: if we wish to show how the major is syllogized from the conclusion and the converse of the minor, let the principal and directly proceeding syllogism be made thus: every B is A, every C is B, therefore every C is A. Then let that major which was the conclusion, namely every C is A, be taken for the reflexive syllogism with the converse of the minor, which is every B is C, and let the conclusion be inferred which was the major of the prior syllogism, thus: every C is A, every B is C, therefore every B is A. Hence if it is necessary to show the major, namely this, that A belongs to every C, let someone show this through B as middle, which in the prior syllogism was the lesser extreme.
Similiter autem ex conclusione et conversa majoris syllogizabitur minor propositio reflexo syllogismo, sic, omne A est B, omne C A, ergo omne C B, quae fuit minor primi syllogismi syllogizata ex conclusione et conversa majoris primi syllogismi: et fit medium A, prius enim sumptum fuit per syllogismum A praedicatum de B quod est e converso sumptum: quia nunc sumitur conversa majoris ubi sumitur B praedicatum de A, quod in terminis convertibilibus nihil est inconveniens. Aliter vero (quam per conclusionem et conversam alterius praemissarum) non est ex se invicem circulariter ostendere sive syllogizare. Sive enim aliud medium sumatur a tribus terminis in primo syllogismo positis, tale medium nihil concludit praemissorum: et tunc non fit reflexio ad principium, unde incipit processus, eo quod tunc nihil sumitur pro medio de numero eorumdem terminorum qui sunt in primo syllogismo: sive etiam accipiatur pro medio quod est de numero horum eorumdem terminorum qui sunt in primo syllogismo: et tunc necesse est, quod alterum solum conversum cum conclusione primi syllogismi accipiatur. Si enim ambo accipiantur (quae erant in priori syllogismo et non conclusio cum conversa alterius) eadem erit sive sequetur conclusio quae prius: et neutro istorum modorum erit reflexio in circulo. In circulari enim syllogismo oportet conclusionem esse diversam a conclusione primi syllogismi: quia aliter non fieret reflexio a conclusione secundi in alterum principiorum primi. Est igitur conveniens et convertibilis inducta diffinitio, et patet ex quibus sic fit circularis syllogismus, et quod aliter fieri non potest. Dicitur autem melius fieri per conclusionem et conversam alterius praemissarum, quam per conclusionem et alteram praemissam, vel quam per conversionem utriusque praemissarum, vel per conversam utriusque praemissarum: quia conclusio sumpta cum una praemissarum, vel conversa utriusque facit dispositionem secundae vel tertiae figurae: tertia autem figura non concludit universalem, et secunda non concludit affirmativam: et ideo necesse est in syllogismo circulari ad concludendam universalem affirmativam praecipue accipi praemissas sub dispositione primae figurae: sic autem se habent conclusio et conversa unius praemissarum: et ideo convenientius dicitur circulus fieri ex conclusione et conversa alterius praemissarum, quam ex conclusione et una praemissarum, vel ex conversione utriusque praemissarum.
Si quis autem objiciens dicat hanc diffinitionem non esse generalem, nec omni circulari syllogismo convenire: quia aliqua circularis ostensio fit per conversam conclusionis, et per alteram propositionem ad syllogizandum conversam alterius: et illi diffinitio non convenit, quia non dicit nisi quod fit per conclusionem et conversam alterius. Dicendum est ad hoc, quod in primo modo primae figurae duae sunt ostensiones circulares. Una est quae per principales praemissas primi syllogismi ostendit conclusionem, et e converso per conclusionem et conversam alterius praemissarum ostendit aliam praemissam: et haec semper sumit conclusionem cum conversione alterius praemissarum, secundum quod directe dicit diffinitio. Alia est quae per conversas propositionum principalium concludit conversam conclusionis principalis, et e converso per ejusdem conclusionis conversam (quae conversa est) et per alteram propositionem conversam concludit conversas propositionum principalium. In hac autem secunda circulatione quae concludit conversam conclusionis principalis, et hujusmodi conclusio accipitur cum conversa alterius praemissarum in reflexo syllogismo circulariter. Huic enim syllogismo est conclusio conversa conclusionis principalis, et praemissa ejus est conversa principalis praemissae, et conversa praemissae ejus est principalis praemissa. Et ex hoc patet quod sicut in prima circulatione syllogizatur una conclusio ex conclusione et conversa alterius propositionis, sic etiam fit in secunda circulatione: et sic patet quod unicuique circulationi secundum se consideratae convenit superius inducta diffinitio.
Likewise from the conclusion and the converse of the major the minor proposition will be syllogized by a reflexive syllogism, thus: every A is B, every C is A, therefore every C is B, which was the minor of the first syllogism, syllogized from the conclusion and the converse of the major of the first syllogism; and A becomes the middle. For previously through the syllogism A had been taken as predicated of B, which is taken conversely, because now the converse of the major is taken, where B is taken as predicated of A; in convertible terms this is not inappropriate. But in another way, other than through the conclusion and the converse of one of the premises, there is no circular showing or syllogizing of things mutually from themselves. For whether another middle is taken apart from the three terms posited in the first syllogism, such a middle concludes none of the premises, and then no reflection is made to the principle from which the process begins, because then nothing from the number of the same terms which are in the first syllogism is taken as middle. Or even if that which is of the number of these same terms in the first syllogism is taken as middle, then necessarily only the converse of one of the two is taken with the conclusion of the first syllogism. For if both are taken, namely those which were in the prior syllogism and not the conclusion with the converse of the other, the conclusion will be the same or will follow as before; and in neither of these two ways will there be reflection in a circle. For in a circular syllogism the conclusion must be different from the conclusion of the first syllogism, because otherwise reflection would not be made from the conclusion of the second syllogism into one of the principles of the first. Therefore the definition introduced is fitting and convertible, and it is clear from what things the circular syllogism is made in this way, and that it cannot be made otherwise. It is said better to be made through the conclusion and the converse of one of the premises than through the conclusion and one premise, or than through the conversion of both premises, or through the converse of both premises, because the conclusion taken with one of the premises, or the converse of both, makes the disposition of the second or third figure. But the third figure does not conclude a universal, and the second does not conclude an affirmative; and therefore in a circular syllogism, for concluding a universal affirmative, it is especially necessary to take premises under the disposition of the first figure. But the conclusion and the converse of one premise are disposed in this way; and therefore the circle is more fittingly said to be made from the conclusion and the converse of one of the premises than from the conclusion and one of the premises, or from the conversion of both premises.
But if someone objecting says that this definition is not general and does not fit every circular syllogism, because some circular showing is made through the converse of the conclusion and through another proposition for syllogizing the converse of the other, and the definition does not fit that because it says only that it is made through the conclusion and the converse of the other, one must say to this that in the first mode of the first figure there are two circular showings. One is that which through the principal premises of the first syllogism shows the conclusion, and conversely through the conclusion and the converse of one of the premises shows another premise; and this always takes the conclusion with the conversion of one of the premises, according as the definition directly says. The other is that which through the converses of the principal propositions concludes the converse of the principal conclusion, and conversely through the converse of that same conclusion, which is the converse, and through the converse of the other proposition, concludes the converses of the principal propositions. But in this second circulation, which concludes the converse of the principal conclusion, a conclusion of this kind is also taken with the converse of one of the premises in the reflexive syllogism circularly. For for this syllogism the conclusion is the converse of the principal conclusion, and its premise is the converse of the principal premise, and the converse of its premise is the principal premise. And from this it is clear that, just as in the first circulation one conclusion is syllogized from the conclusion and the converse of one proposition, so also it happens in the second circulation; and thus it is clear that the definition introduced above fits each circulation considered in itself.
Ex jam dictis autem facile est solvere id quod hic quaeri solet, scilicet cum (sicut dictum est) fiat syllogismus circularis, utrum unusquisque circulorum secundum se circulus sit, aut sint omnes unus circulus; neutrum enim horum videtur esse; quia nec primo modo nec secundo fit reditus in idem a quo incipit decursus syllogismi. Patet enim ex hoc verum esse, quod non sit in idem reditus: sed cum fiant sex syllogismi juxta primum modum primae figurae (ut ostendetur postea), isti sex duos faciunt circulos.
CAPUT II.
Qualiter fit circularis syllogismus in prima figura universalis affirmativus et universalis negativus, et qualiter particularis affirmativus et negativus.
Ad sciendum autem qualiter in singulis figuris circularis sit syllogismus, primo sciendum est quod in his terminis vel propositionibus quae non convertuntur sive quae sunt in terminis non convertibilibus si fiat syllogismus circularis, oportet quod procedat ex indemonstrata propositione: non est enim demonstrare sive ostendere circulariter per hos terminos qui non sunt convertibiles, quod medio insit tertium (hoc est minus extremum quod tertium est in dispositione figurae primae) per conversionem minoris: quia tales termini non convertuntur: et ideo subjectum non potest fieri praedicatum, et e converso universaliter accepta minori propositione: et eadem de causa non est accipere in majori propositione, qualiter primum, id est, majus extremum insit medio, sicut subjectum est contentum in praedicato.
In his vero terminis qui convertuntur et pares sunt invicem, est omnia demonstrare sive ostendere per se invicem circulariter, sicut patet in his terminis A B C in primo modo primae figurae, et primo in principali syllogismo ostendatur haec conclusio, omne C A per B medium, sic, omne B A, et omne C B, ergo omne C A. Et in reflexo syllogismo rursum ostendatur major quae est, omne B A per conclusionem inductam et per B C minorem propositionem conversam, sic, omne C A, omne B C, ergo omne B A, quae fuit major in principali syllogismo. Similiter autem circulariter reflectendo ostendetur minor quae est C B propositio per conclusionem et majorem conversam, sic, omne A B, quae est conversa majoris: omne C A, quae est conclusio: sequitur quod omne C B, quae fuit minor principalis syllogismi. Iste igitur est unus circulus tribus syllogismis conclusus.
Oportet autem et conversas harum propositionum per alium circulum demonstrare: et ideo oportet demonstrare et concludere et B C propositionem (quae est conversa minoris) et B A propositionem (quae est conversa majoris) per circulum demonstrare sive concludere. Nam his solis usi sumus in praecedenti circulatione in demonstratis: eo quod illae conclusae non fuerunt per aliquem syllogismum circuli prius positi. Dicamus igitur quod si sumatur minor principalis syllogismi quae est B omni C inesse, sive quod omne C est B, et jungatur ei A C propositio quae est conversa conclusionis, sic omne A C, erit syllogismus concludens conversam majoris, hoc est, B ad A universaliter praedicari, sic, omne C B, omne A C, ergo omne A B. Rursum autem si sumatur C omni A inesse, quod est conversa conclusionis principalis: et sumatur A omni B inesse, quod est conversa majoris: necesse est sequi C inesse omni B, quod est conversa minoris.
From what has now been said it is easy to solve what is commonly asked here, namely, when a circular syllogism is made, as has been said, whether each of the circles is a circle in itself, or whether all are one circle. For neither of these seems to be the case, since neither in the first way nor in the second is there a return into the same thing from which the descent of the syllogism begins. From this it is clear that it is true that there is no return into the same thing; but since six syllogisms are made next to the first mode of the first figure, as will be shown afterwards, those six make two circles.
CHAPTER II.
How a circular syllogism is made in the first figure, both universal affirmative and universal negative, and how a particular affirmative and negative is made.
For knowing how a circular syllogism is made in the individual figures, one must first know that, in those terms or propositions which are not converted, or which are in non-convertible terms, if a circular syllogism is made, it must proceed from an indemonstrate proposition. For it is not possible to demonstrate or show circularly through these terms which are not convertible that a third thing belongs to the middle, that is, the lesser extreme which is third in the disposition of the first figure, through conversion of the minor; because such terms are not converted, and therefore the subject cannot become predicate, and conversely, when the minor proposition is accepted universally. For the same reason it is not possible to take in the major proposition how the first, that is, the greater extreme, belongs to the middle as subject is contained in predicate.
But in those terms which are converted and are equal to one another, it is possible to demonstrate or show all things through one another circularly, as is clear in these terms A B C in the first mode of the first figure. And first, in the principal syllogism, let this conclusion be shown, every C is A, through B as middle, thus: every B is A, and every C is B, therefore every C is A. And in the reflexive syllogism, again let the major be shown, which is every B is A, through the conclusion brought in and through the converted minor proposition B C, thus: every C is A, every B is C, therefore every B is A, which was the major in the principal syllogism. Likewise, by reflecting circularly, the minor, which is the proposition C B, will be shown through the conclusion and the converted major, thus: every A is B, which is the converse of the major; every C is A, which is the conclusion; it follows that every C is B, which was the minor of the principal syllogism. Therefore this is one circle completed by three syllogisms.
But it is also necessary to demonstrate the converses of these propositions through another circle; and therefore it is necessary also to demonstrate and conclude both the B C proposition, which is the converse of the minor, and the B A proposition, which is the converse of the major, through the circle. For in the preceding circulation we used only these in the things demonstrated, because those had not been concluded through any syllogism of the circle previously posited. Therefore let us say that if the minor of the principal syllogism is taken, which is that B belongs to every C, or that every C is B, and A C, the proposition which is the converse of the conclusion, is joined to it, thus every A is C, there will be a syllogism concluding the converse of the major, that is, B being predicated universally of A, thus: every C is B, every A is C, therefore every A is B. Again, if C is taken to belong to every A, which is the converse of the principal conclusion, and A is taken to belong to every B, which is the converse of the major, it is necessary that C follow as belonging to every B, which is the converse of the minor.
In utrisque autem his syllogismis circularibus sumpta est A C propositio (quae est conversa conclusionis) indemonstrata: eo quod per nullum dictorum syllogismorum est conclusa: omnes enim aliae propositiones et conversae earum probatae sunt et conclusae: propter quod si hanc circulari syllogismo demonstraverimus sive concluserimus, omnes erunt approbatae sive conclusae per se invicem. Si ergo sumatur C omni B inesse, quae est conversa minoris principalis syllogismi: et sumatur conversa majoris, scilicet quod B insit omni A, et utraeque dictae propositiones sumantur demonstratae et conclusae per circularem syllogismum in secundo circulo positum, sequitur quod C necesse est inesse A, quae est conversa conclusionis principalis syllogismi.
Manifestum est igitur ex his quae dicta sunt quoniam in solis his terminis qui convertuntur, circulo et per se invicem contingit fieri demonstrationes sive conclusiones. In aliis autem non convertibilibus fiunt ostensiones, quemadmodum prius in anteriori libro dictum est. In his autem quae convertibilia sunt eodem sumpto, hoc est, eadem conclusione quae demonstratur in primo syllogismo, accidit uti pro propositione ad demonstrationem alterius conclusionis, quae est conclusio reflexi syllogismi in circulum: cujus signum est: quia in syllogismo reflexo C de B quae est conversa minoris, et B de A quae est conversa majoris, monstratur sumpto C de A dici, conversa scilicet conclusionis, C autem de A (conversa B conclusionis) ostenditur per has propositiones acceptas conversas: propter quod conclusione sumpta in circulari syllogismo pro praemissa utimur ad demonstrationem praemissarum.
In privativis autem syllogismis universalibus primae figurae circularibus hoc modo conclusionis et propositionum fit demonstratio ex se invicem. Et dico privativis syllogismis: quia licet unus solus sit in prima figura privativus et universalis syllogismus: tamen quia duobus circulis clauditur, plures ad hoc in circulari syllogismo necesse est fieri syllogismos. Fiat autem syllogismus principalis sic: accipiatur enim minor universalis affirmativa, ita quod B dicatur omni C inesse: et accipiatur major negativa, ita quod dicatur A nulli B inesse: conclusio erit principalis syllogismi, quoniam A nulli C inerit, sic, nullum B A, omne C B, ergo nullum C A. Si ergo rursum in reflexo syllogismo oporteat concludere majorem principalis syllogismi (hanc scilicet) quod A nulli insit B, sive quod nullum B est A, quod prius in principali syllogismo sumptum erat pro majori: tunc sumetur conclusio principalis syllogismi, haec scilicet quod A nulli C inest, et adjungetur conversa minori, haec scilicet quod C inest omni B (sic enim e converso sumitur minor propositio), et concluditur quod nullum B A, sic, nullum C A, omne B C, ergo nullum B A, quae fuit major principalis syllogismi.
Si autem oportet concludere circulari syllogismo minorem principalis syllogismi, hanc scilicet, quoniam B inest omni C, sive quod omne C est B, non similiter ut prius in affirmativo syllogismo convertenda est A B major propositio: nam eadem quantum ad qualitatem est propositio major quae dicit, quoniam A nulli inest B, et sua conversa quae dicit, quod nulli A inest B, ambae enim sunt negativae: et affirmativa (quae est minor concludenda) non sequitur ex negativa: et ideo majoris conversa reducenda est in affirmativam sic: major enim est, nullum B est A, et convertenda in hanc, cui A nulli inest, huic omni inest B, haec enim est affirmativa, scilicet C nulli inest A quae fuit conclusio: ergo C omni inest B quae fuit minor in syllogismo principali.
But in both of these circular syllogisms the proposition A C, which is the converse of the conclusion, has been taken as indemonstrate, because it has been concluded through none of the syllogisms mentioned. For all the other propositions and their converses have been proved and concluded. On account of this, if we shall have demonstrated or concluded this one by a circular syllogism, all will have been approved or concluded through one another. If, therefore, C is taken to belong to every B, which is the converse of the minor of the principal syllogism, and the converse of the major is taken, namely that B belongs to every A, and both of the said propositions are taken as demonstrated and concluded through the circular syllogism posited in the second circle, it follows that C must belong to A, which is the converse of the conclusion of the principal syllogism.
Therefore it is manifest from what has been said that only in these terms which are converted can demonstrations or conclusions happen through a circle and mutually through themselves. But in other non-convertible terms, showings are made as was previously said in the earlier book. In those things which are convertible, when the same thing is taken, that is, the same conclusion which is demonstrated in the first syllogism, it happens that it is used as a proposition for demonstrating another conclusion, which is the conclusion of the reflexive syllogism into the circle. A sign of this is that in the reflexive syllogism C of B, which is the converse of the minor, and B of A, which is the converse of the major, is shown by taking C to be said of A, namely the converse of the conclusion; but C of A, the converse of B as conclusion, is shown through these propositions accepted as converted. On account of this, in the circular syllogism we use the conclusion taken as a premise for the demonstration of the premises.
But in universal privative syllogisms of the first figure, circular demonstrations from one another are made in this way from conclusion and propositions. And I say in privative syllogisms because, although in the first figure there is only one privative and universal syllogism, nevertheless, because it is closed by two circles, it is necessary for several syllogisms to be made for this in the circular syllogism. Let the principal syllogism be made thus: let the minor universal affirmative be accepted, so that B is said to belong to every C; and let the negative major be accepted, so that A is said to belong to no B. The conclusion of the principal syllogism will be that A will belong to no C, thus: no B is A, every C is B, therefore no C is A. Therefore if again in the reflexive syllogism it is necessary to conclude the major of the principal syllogism, namely this, that A belongs to no B, or that no B is A, which previously in the principal syllogism had been taken as major, then the conclusion of the principal syllogism will be taken, namely this, that A belongs to no C, and the converted minor will be joined, namely this, that C belongs to every B, for in this way the minor proposition is taken conversely, and it is concluded that no B is A, thus: no C is A, every B is C, therefore no B is A, which was the major of the principal syllogism.
But if it is necessary to conclude by circular syllogism the minor of the principal syllogism, namely this, that B belongs to every C, or that every C is B, the major proposition A B is not to be converted in the same way as before in the affirmative syllogism. For the major proposition, which says that A belongs to no B, and its converse, which says that B belongs to no A, are the same with respect to quality, since both are negative; and the affirmative, which is the minor to be concluded, does not follow from a negative. Therefore the converse of the major must be reduced to an affirmative in this way: for the major is, no B is A, and it is to be converted into this, that to whatever A belongs in no way, B belongs to it universally. For this is affirmative: namely, C belongs to no A, which was the conclusion; therefore C belongs to every B, which was the minor in the principal syllogism.
Propter quod cum tria sint in syllogismo principali, unumquodque illorum conclusio facta est: tria enim sunt major et minor et conclusio: circulo enim demonstrare (sicut diximus) hoc est aliquem sumentem conclusionem principalis syllogismi cum conversa alterius praemissarum reliquam (quae non convertitur) syllogizando in reflexo syllogismo concludere.
Si autem quaeritur, quare juxta modum universalem negativum primae figurae non fiunt sex syllogismi, sicut circa modum affirmativum: hic enim non syllogizantur nisi praemissae tantum, et non syllogizatur conclusio earum. Dicendum quod hoc ideo est, quia in negativo universali syllogismo primae figurae non fit nisi una circulatio, quamvis juxta primum modum fuerant duae circulationes: ex conversis enim praemissarum potest ostendi conversa conclusionis: nec ex conversa conclusionis et minori potest ostendi conversa majoris, eo quod utrobique hic et ibi accipitur minor negativa, quod in prima figura esse non potest: nec ex conversa conclusionis et majori concluditur conversa minoris, quia utraque est negativa, et ex utraque negativa non fit syllogismus in aliqua figura.
Sed adhuc forte quaeret aliquis, quare non transponuntur illae negativae in affirmativas, ita quod ex illis syllogizaretur, sicut syllogizatur ad ostendendam minorem principalis syllogismi. Et dicendum est ad hoc, quod transpositio talis in illis conversis esset inutilis, et nihil faceret ad propositum: quod sic patet: fiat enim syllogismus principalis sic, nullum B A, omne C B, ergo nullum C A. Si debeat ostendi conversa conclusionis, haec scilicet, nullum A C per conversas praemissarum. Accipiatur ergo conversa minoris, haec scilicet, omne B est C, et deinde jungatur conversa majoris transposita in affirmativam, sic, cui nulli inest A illi omni inest B, et patet quod hoc nihil est ad propositum: ex his enim duabus (omne B est C cui nulli inest A, omni illi inest B) non potest syllogizari, quod C nulli insit A. Similiter autem est et in aliis conversis concludendis: et haec quidem est veritas.
Tamen quidam non subtiliter ista perspicientes, concedunt quod conversae principalium propositionum syllogizari possint per transpositionem negativam in affirmativas, eo modo quo dictum est. Est tamen quaestio de hujusmodi conversione: quia videtur non esse generalis hujusmodi conversio, et non tenere nisi in materia contingenti. Adhuc autem ubi vera conversio est, si convertens est vera, est etiam vera ejus conversa: sed in ista conversione non est sic: si enim sic dicam, nullus homo est asinus, haec vera est: et si convertatur, sic, cui nulli inest asinus, illud omne est homo, falsum erit: quia nulli ligno inest asinus: et tamen non omne lignum est homo.
Dicendum autem ad hoc, quod non vera est conversio, sed translatio conversae negativae in affirmativam: unde negativa primo est convertenda, et postea conversa transferenda, sic, nullum B est A in hanc convertatur, nullum A B et postea transferatur, sic, nulli cui inest A illi omni inest B. Sicut autem in affirmativa propositione termini sunt convertibiles, ita et in negativa termini quantum possunt debent accedere ad convertibilitatem, ita quod sint contradicentes negative et immediati, ita quod necesse sit alterum inesse sicut in contradictoriis: sicut enim dicitur, nullum aequale est inaequale: haec enim si transmutaretur in affirmativa, affirmativa erit vera sicut negativa, sic, cui nulli inest inaequale, ei omni inest aequale: unde talis conversio non est nisi quaedam coarctatio circa majorem propositionem in syllogismo negativo: quae coarctatio fit ad terminos convertibiles, vel ad convertibilitatem accedentes, et oppositionem immediatam quae est per modum contradictionis.
On account of this, since there are three things in the principal syllogism, each one of them has been made a conclusion. For there are three: major, minor, and conclusion. For to demonstrate in a circle, as we have said, is for someone, taking the conclusion of the principal syllogism with the converse of one of the premises, to conclude in a reflexive syllogism the remaining one, which is not converted, by syllogizing.
But if it is asked why next to the universal negative mode of the first figure six syllogisms are not made, as about the affirmative mode, for here only the premises are syllogized and their conclusion is not syllogized, one must say that this is so because in the negative universal syllogism of the first figure there is only one circulation, although next to the first mode there had been two circulations. For from the converses of the premises the converse of the conclusion can be shown; but from the converse of the conclusion and the minor, the converse of the major cannot be shown, because both here and there a negative minor is accepted, which cannot be in the first figure; nor is the converse of the minor concluded from the converse of the conclusion and the major, because both are negative, and from two negatives no syllogism is made in any figure.
But perhaps someone will still ask why those negatives are not transposed into affirmatives, so that something would be syllogized from them as is syllogized for showing the minor of the principal syllogism. One must say to this that such a transposition in those converses would be useless and would contribute nothing to the proposed point. This is clear in this way: let the principal syllogism be made thus: no B is A, every C is B, therefore no C is A. If the converse of the conclusion is to be shown, namely this, no A is C, through the converses of the premises, let the converse of the minor be accepted, namely this, every B is C, and then let the converse of the major, transposed into an affirmative, be joined thus: to whatever A belongs in no way, B belongs universally to that thing. And it is clear that this contributes nothing to the proposed point. For from these two, every B is C, and to whatever A belongs in no way, to all that B belongs, it cannot be syllogized that C belongs to no A. Likewise it is so also in the other converses to be concluded; and this indeed is the truth.
Nevertheless, certain people, not considering these matters subtly, concede that the converses of the principal propositions can be syllogized through the transposition of a negative into affirmatives, in the way that has been stated. Yet there is a question about conversion of this kind, because such conversion seems not to be general and not to hold except in contingent matter. Again, where there is true conversion, if the converting proposition is true, its converse is also true; but in this conversion it is not so. For if I say thus, no human being is a donkey, this is true; and if it is converted thus, to whatever donkey belongs in no way, that whole thing is human, it will be false, because donkey belongs to no piece of wood, and yet not every piece of wood is a human being.
But one must say to this that it is not a true conversion, but a translation of a converted negative into an affirmative. Hence the negative is first to be converted, and afterwards the converted proposition is to be translated; thus, no B is A is converted into this, no A is B, and afterwards it is translated thus: to none to which A belongs, to all that B belongs. But just as in an affirmative proposition the terms are convertible, so also in a negative proposition the terms ought, as far as they can, to approach convertibility, so that they are negatively contradicting and immediate, so that necessarily one of them belongs, as in contradictories. For example, it is said: no equal thing is unequal. If this were transmuted into an affirmative, the affirmative would be true just as the negative is: to whatever unequal does not belong in any way, to every such thing equal belongs. Hence such conversion is only a certain narrowing about the major proposition in a negative syllogism; this narrowing is made to convertible terms, or to terms approaching convertibility, and to immediate opposition, which is by way of contradiction.
Posset tamen dici quod in tali conversione non fit coarctatio ad terminos convertibiles: sed cum dicitur, cui nulli inest A illi omni inest B, per hoc quod dicitur cui, intelligendum est quid particulare a quo universaliter removetur A, et cui universaliter insit B; et illud oportet necessario esse C, quia, sicut dictum est in hujus tractatus antecedentibus, in circulari ostensione non debet a sumi terminus novus extra praeacceptos in principali syllogismo: quia aliter non esset ostensio circularis: propter quod per id quod dicitur cui, per virtutem ostensionis circularis oportet intelligere C, et sic est propositio vera sic, alicui sicut A cui nulli inest A, omni illi inest B.
Si autem quaeritur de syllogismo qui fit ex hujusmodi conversa sic, cui nulli inest A, illi omni inest B, sed C est cui nulli inest A, ergo omni C inest B. Talis enim conversio non videtur esse syllogismus: quia major est particularis respectu conclusionis: propter hoc quod dico cui, particulariter supponit pro C, et etiam propter hoc quod minor est negativa: quorum utrumque est inconveniens in prima figura. Adhuc quia conclusio est affirmativa, et una praemissarum negativa: quod in nulla figura fieri potest in forma syllogistica. Adhuc si dicatur esse syllogismus primae figurae, quaeritur in quo modo sit, et non erit assignare. Si autem diceretur syllogismus esse in tertia figura, hoc stare non potest, quia tunc oporteret conclusionem converti sicut majorem, et hoc fieri non debet: quia syllogizatur ex conclusione et altera praemissarum conversa. Adhuc in tertia figura non syllogizatur universalis: hic autem concluditur universalis propositio.
Ad hoc autem dicendum quod talis syllogismus est quidem in prima figura, sed non in aliquo modorum primae figurae, eo quod non directe ostensivus: nec habet virtutem inferendi ab eo quod est dici de omni et dici de nullo, sed virtutem inferendi quasi sicut conditionalis quae talis est, quod si ab aliquo removeatur unum, quod continue inferatur reliquum, ut si non est homo, est non homo. Unde totus syllogismus ille est quasi hypotheticus, et inferens a positione antecedentis: nec major est particularis respectu conclusionis, sed est in aeque ei: eo quod hoc quod dico cui, pro C supponit, quod idem C subjicitur in conclusione. Nec etiam est inconveniens in tali syllogismo a positione antecedentis inferre ex minori negativa: quia negatio non cadit super totum medium: quinimo pars ipsius medii negatio est: nec inconveniens est ex sic et taliter negativa inferre affirmativam.
In particularibus autem syllogismis duobus primae figurae, scilicet tertio et quarto hic communiter notandum, quod universalem propositionem talis syllogismi (majorem scilicet) non est demonstrare sive concludere per circulum: quia si concluderetur, non posset concludi nisi per conclusionem quae particularis est, et per conversam minoris, quae iterum est particularis, et ex utraque particulari non fit syllogismus in aliqua figura: sed particularis circulo demonstratur. Quoniam autem non est demonstrare universalem propositionem in particulari syllogismo, sic probatur. Universalis enim non demonstratur sive concluditur nisi per universales utrasque propositiones: sed conclusio (ex qua proceditur in reflexo syllogismo) non est universalis. Dictum autem est, quod in reflexo syllogismo oporteret fieri ostensionem ex conclusione et altera praemissarum conversa.
Amplius omnino et universaliter in nulla figura fit syllogismus conversa minori propositione quae particularis est affirmativa: quia si illa conversa accipiatur cum conclusione, quae et ipsa est particularis, utraeque propositiones praemissae fiunt particulares: et ex talibus in nulla figura est syllogizare. Particularem autem propositionem in syllogismo particulari positam, est ostendere et concludere circulariter. Ostendatur, hoc est, concludatur A majus extremum de aliquo C minori particulariter per medium B, sic, omne B A, quoddam C B, ergo quoddam C A. Si ergo sumatur major propositio conversa, ita dicatur B omni A inesse, et conclusio maneat principalis syllogismi, hujus scilicet, quoddam C A, concluditur minor prioris syllogismi, scilicet quod B alicui C inerit, sic, omne A B, quoddam C A, ergo quoddam C B. Sic enim fit prima figura, et A fit medium quod in principali syllogismo fuit majus extremum.
Still it could be said that in such conversion there is no narrowing to convertible terms. Rather, when it is said, to whatever A belongs in no way, to all that B belongs, through the word whatever one must understand some particular thing from which A is universally removed and to which B universally belongs; and that thing must necessarily be C, because, as has been said in what precedes in this tractate, in circular showing a new term outside those already accepted in the principal syllogism must not be taken, because otherwise it would not be a circular showing. On account of this, through that which is said, whatever, by the force of circular showing one must understand C; and so the proposition is true thus: to something such as A, to which A belongs in no way, B belongs universally.
But if one asks about the syllogism which is made from a converse of this sort thus, to whatever A belongs in no way, to all that B belongs; but C is that to which A belongs in no way; therefore B belongs to every C. Such a conversion does not seem to be a syllogism, because the major is particular with respect to the conclusion, since what I call whatever stands particularly for C, and also because the minor is negative; each of these is inappropriate in the first figure. Again, because the conclusion is affirmative and one of the premises negative, which cannot occur in any figure in syllogistic form. Again, if it is said to be a syllogism of the first figure, it is asked in which mode it is, and it will not be possible to assign one. But if it were said to be a syllogism in the third figure, this cannot stand, because then the conclusion would have to be converted just as the major is, and this should not be done, because it is syllogized from the conclusion and the converse of one of the premises. Again, in the third figure a universal is not syllogized; but here a universal proposition is concluded.
To this, however, one must say that such a syllogism is indeed in the first figure, but not in any of the modes of the first figure, because it is not directly ostensive. Nor does it have the power of inferring from that which is "to be said of every" and "to be said of none," but the power of inferring almost as a conditional does, which is such that if one thing is removed from something, what remains is immediately inferred, as, if it is not a human being, it is non-human. Hence that whole syllogism is almost hypothetical, and it infers from the positing of the antecedent. Nor is the major particular with respect to the conclusion, but is equal to it, because what I call whatever stands for C, since that same C is subjected in the conclusion. Nor is it inappropriate in such a syllogism to infer from the positing of the antecedent from a negative minor, because the negation does not fall upon the whole middle; rather the negation is part of the middle itself. Nor is it inappropriate to infer an affirmative from a negative in this way and in such a manner.
In the two particular syllogisms of the first figure, namely the third and fourth, this must be noted in common here: the universal proposition of such a syllogism, namely the major, cannot be demonstrated or concluded through a circle, because if it were concluded, it could be concluded only through the conclusion, which is particular, and through the converse of the minor, which again is particular; and from two particular premises no syllogism is made in any figure. But the particular is demonstrated by a circle. That it is not possible to demonstrate the universal proposition in a particular syllogism is proved thus. A universal is not demonstrated or concluded except through two universal propositions; but the conclusion, from which the process proceeds in the reflexive syllogism, is not universal. And it has been said that in a reflexive syllogism a showing must be made from the conclusion and the converse of one of the premises.
Moreover, absolutely and universally, in no figure is a syllogism made by converting the minor proposition which is a particular affirmative, because if that converse is accepted with the conclusion, which itself is particular, both premise-propositions become particular; and from such propositions there is no syllogizing in any figure. But a particular proposition posited in a particular syllogism can be shown and concluded circularly. Let A, the greater extreme, be shown, that is, concluded, particularly of some C, the minor, through the middle B, thus: every B is A; some C is B; therefore some C is A. Therefore, if the converted major proposition is taken, so that B is said to belong to every A, and the conclusion of the principal syllogism remains, namely this, some C is A, the minor of the prior syllogism is concluded, namely that B will belong to some C, thus: every A is B; some C is A; therefore some C is B. For in this way the first figure is made, and A becomes the middle, which in the principal syllogism was the greater extreme.
Si autem fit privativus syllogismus particularis sicut in quarto modo primae figurae, sic, nullum B A, quoddam C B, ergo quoddam C non est A. Universalem majorem negativam non est demonstrare per circulum: propter hoc quod prius dictum est in tertio modo primae: quia eaedem causae impediunt et hic et ibi. Sed particularem propositionem affirmativam minorem est vel contingit demonstrare per circulum. Hoc autem fit ex conclusione et conversa majoris, si similiter convertatur major propositio A B, sicut in universali syllogismo negativo dictum est, sic scilicet, ut cui alicui non inest A, huic alicui inest B, et syllogizatur sic, cui alicui non inest A, illi alicui inest B, sed C alicui non inest A, ergo C alicui inest B. Nam aliter non potest fieri syllogismus: eo quod negativa est particularis propositio, quae est conclusio syllogismi principalis, ex qua procedit reflexus syllogismus, quae cum alia particulari nihil infert: et cum majori conversa negativa ex utraque negativa syllogizaret: quod fieri non potest in aliqua trium figurarum: sic igitur convertenda est major, et fiat circulus.
De his tamen particularibus syllogismis posset aliquis dubitare: quia in affirmativo syllogismo (qui est tertius primae) non docetur syllogizari nisi minor tantum, non major, nec conversa majoris, nec conversa conclusionis, et quod major tam in syllogismo affirmativo quam in negativo concludi non potest, satis in praedictis ostensum est: quia utraque est universalis: et universalis concludi non potest: conversa autem conclusionis concludi non posset, nisi per conversionem majoris, quae in affirmativo non convertitur nisi in particularem: et tunc illa cum conclusione (quae etiam particularis est) sumpta non facit syllogismum. In negativo autem quando convertitur major, et sumitur cum conclusione, concludit minorem: ad concludendam autem majorem vel conversam ejus oporteret, quod particularis vel minor, vel conclusio, vel earumdem conversae, efficerentur majores in prima figura: quod esse non potest. Et per idem patet quod in universali modo primae perfectus est circulus, in modo autem negativo universali imperfectus. Sed in particularibus modis penitus est circulus diminutus. Adhuc autem circa quartum modum primae figurae negativum, si praeinducto modo reflectatur in circulum, videtur syllogismus fieri ex particularibus. Adhuc videtur concludere conclusionum negativam ex affirmativis: quorum neutrum fieri debet.
Sed ad hoc (sicut superius dictum est) responderi habet: quia pro certo cum dicitur, cui alicui non inest A, illi alicui inest B, propositio ista virtutem habet hypotheticae propositionis conditionalis: et virtus inferendi in tali syllogismo tota concluditur in majori: et ideo non infert per dici de omni et dici de nullo, sed infert a positione antecedentis: et quia concludit minorem quae est affirmativa, oportet quod transferatur in affirmativam, sicut superius dictum est: et tamen quia hic est particularis syllogismus, hic transferenda in terminos particulares. Superius autem in universali negativo transfertur in terminos non particulares, sed universales: et sub B non potest accipi nisi C per virtutem syllogismi circularis, in quo non debet sumi novus terminus. Nec est inconveniens ex tali particulari concludere et facere eam majorem: quia (sicut dictum est) non infert per dici de omni vel de nullo, sed ex virtute hypothesis a positione antecedentis. Et debent isti syllogismi sic formari: universalis quidem sicut supra dictum est: particularis autem sic, cui alicui ut C non inest A, illi alicui inest B, sed C est cui alicui non inest A, ergo C alicui inest B.
But if a particular privative syllogism is made as in the fourth mode of the first figure, thus: no B is A; some C is B; therefore some C is not A, the universal negative major cannot be demonstrated through a circle, on account of what was said earlier in the third mode of the first figure, because the same causes impede here and there. But the particular affirmative proposition, the minor, can or does happen to be demonstrated through a circle. This is done from the conclusion and the converse of the major, if the major proposition A B is similarly converted as was said in the universal negative syllogism, namely in this way: to whatever particular thing A does not belong, to this particular thing B belongs; and it is syllogized thus: to whatever particular thing A does not belong, to that particular thing B belongs; but C is something to which A does not belong; therefore B belongs to some C. For otherwise no syllogism can be made, because the proposition which is the conclusion of the principal syllogism, from which the reflexive syllogism proceeds, is a particular negative, and together with another particular it infers nothing; and with the converted negative major it would syllogize from two negatives, which cannot be done in any of the three figures. Therefore the major must be converted in this way, and the circle is made.
Yet concerning these particular syllogisms someone could doubt, because in the affirmative syllogism, which is the third of the first figure, only the minor is taught to be syllogized, not the major, nor the converse of the major, nor the converse of the conclusion; and that the major, both in an affirmative syllogism and in a negative one, cannot be concluded has been sufficiently shown in what was said before, because each is universal, and a universal cannot be concluded. But the converse of the conclusion could not be concluded except through conversion of the major, which in an affirmative syllogism is converted only into a particular; and then that proposition, taken with the conclusion, which is also particular, does not make a syllogism. But in the negative, when the major is converted and is taken with the conclusion, it concludes the minor; yet for concluding the major or its converse, it would be necessary that the particular, whether the minor or the conclusion, or their converses, become majors in the first figure, which cannot be. And through the same point it is clear that in the universal mode of the first figure the circle is perfect, but in the universal negative mode it is imperfect. But in the particular modes the circle is entirely diminished. Moreover, concerning the fourth negative mode of the first figure, if it is reflected into a circle in the way introduced before, the syllogism seems to be made from particulars. Again, it seems to conclude a negative conclusion from affirmatives, neither of which ought to be done.
But to this, as was said above, one must reply, because certainly when it is said, to whatever particular thing A does not belong, to that particular thing B belongs, that proposition has the power of a hypothetical conditional proposition; and the whole power of inferring in such a syllogism is concluded in the major. Therefore it does not infer through "being said of every" and "being said of none," but infers from the positing of the antecedent. And because it concludes the minor, which is affirmative, it must be transferred into an affirmative, as was said above. Nevertheless, because here there is a particular syllogism, it must here be transferred into particular terms. But above, in the universal negative, it is transferred into terms not particular but universal; and under B nothing can be taken except C through the force of the circular syllogism, in which a new term must not be taken. Nor is it inappropriate to conclude from such a particular and to make it the major, because, as has been said, it does not infer through "being said of every" or "of none," but from the power of a hypothesis by positing the antecedent. And these syllogisms should be formed thus: the universal indeed as was said above; the particular in this way: to whatever particular thing, as C, A does not belong, to that particular thing B belongs; but C is that to which A does not belong; therefore B belongs to some C.
Si autem quaeritur quare syllogismi sic formati reducuntur ad formam ostensivorum syllogismorum in prima figura? Dicendum quod ille qui fit juxta modum universalem (secundum scilicet primae figurae) sic potest reduci, omne C non ens A est B, sed C est C non ens A, ergo omne C est B. Particularis autem qui fit juxta quartum modum primae figurae, reduci sic habet in categoricum et simplicem et ostensivum syllogismum, omne C non ens A est B, aliquod C est C non ens A, ergo aliquod C est B: quia cum conclusio affirmativa sit, oportet tam conversam majoris quam conclusionem transferre in affirmativam, et sic syllogizare. Ex his igitur patet qualiter fit circulus in prima figura.
CAPUT III.
Qualiter circulus fit in secunda figura.
In secunda autem figura affirmativam propositionem non est ostendere circulo, sed privativam in secunda figura est per circulum ostendere. Praedicativa quidem sive affirmativa in secunda figura non ostenditur hac de causa: quia cum per circulum ostendere sit, cum conclusione et conversa unius alteram syllogizare, oportet quod altera semper esset negativa: quia conclusio in secunda figura semper est negativa: sed ex altera negativa non sequitur conclusio affirmativa: ergo affirmativam non contingit concludere in secunda figura: affirmativa enim non sequitur nisi ex utraque affirmativa.
Contra hoc tamen esse videtur: quia in sequentibus ostendetur, quod ex conversa negativae transposita in affirmativam, syllogizatur affirmativa. Adhuc in modis negativis primae figurae syllogizatur affirmativa, in quibus non est utraque negativa: videtur igitur, quod hoc etiam hic fieri possit in secunda figura. Adhuc autem in primo modo secundae videtur syllogizari affirmativa ex negativa majori transposita in affirmativam, sic, cui nulli inest B illi omni inest A, C autem est cui nulli inest A, ergo omni C inest B.
Ad haec autem et similia dicendum quod non fit circularis ostensio, nisi ubi conclusio sumitur cum conversa alterius praemissarum ad concludendam reliquam: et hoc modo non potest syllogizari in secunda figura affirmativa secundum circulum. Si autem conversa negativae transferatur in affirmativam, nihil faciet hoc modo ad circularem ostensionem: in sequentibus in tertio modo syllogizabitur affirmativa, non ex conclusione et conversa majoris transposita in affirmativam, sed ex ipsa majori sic transposita in qua est tota virtus illationis, sicut dictum est: sed sic syllogizare non est circulo ostendere. Et licet sic syllogizetur affirmativa syllogismorum universalium sicut objectum est: non tamen fit hoc secundum circulum, sicut patet ex diffinitione circularis ostensionis, cum ibi sumitur conclusio cum altera praemissarum conversa: quia si sumeretur conversa negativae transposita in affirmativam, non esset ad propositionum concludendam conclusionem circulariter: sed vel sumitur ipsa propositio translata in affirmativam cum conclusione vel cum conversa conclusionis. Ad id autem quod objicitur de negativis primae figurae, dicendum quod aliud est de negativis primae figurae, et aliud de negativis secundae. In prima enim figura ex conclusione et conversa alterius praemissae syllogizatur reliqua. Sed in secunda figura hoc nihil facit ad propositum conclusio et conversa negativae: quia ex conclusione negativae non syllogizatur affirmativa. Causa enim praecisa quare hic non syllogizatur affirmativa, haec est: quia non fit circularis ostensio nisi ex conclusione et altera praemissarum conversa: ex conclusione autem et conversa non fit hic syllogismus ad propositum in secunda figura. Sic igitur patet quod circularis ostensio non fit de affirmativa propositione in secunda figura.
But if it is asked why syllogisms formed in this way are reduced to the form of ostensive syllogisms in the first figure, one must say that the one which is made next to the universal mode, namely the second of the first figure, can be reduced thus: every C which is a non-being of A is B; but C is a C which is a non-being of A; therefore every C is B. But the particular one which is made next to the fourth mode of the first figure has to be reduced in this way into a categorical, simple, and ostensive syllogism: every C which is a non-being of A is B; some C is a C which is a non-being of A; therefore some C is B. For since the conclusion is affirmative, it is necessary to transfer both the converse of the major and the conclusion into an affirmative, and so to syllogize. From these things, therefore, it is clear how a circle is made in the first figure.
CHAPTER III.
How a circle is made in the second figure.
In the second figure, however, it is not possible to show an affirmative proposition by a circle, but it is possible to show a privative proposition in the second figure through a circle. The predicative, or affirmative, proposition in the second figure is not shown for this reason: because since to show through a circle is to syllogize one premise from the conclusion and the converse of another, it would be necessary that one premise always be negative, because the conclusion in the second figure is always negative. But an affirmative conclusion does not follow from one negative premise; therefore it is not possible to conclude an affirmative in the second figure. For an affirmative follows only from two affirmatives.
Nevertheless, the contrary seems to be the case, because in what follows it will be shown that from the converse of a negative transposed into an affirmative an affirmative is syllogized. Again, in the negative modes of the first figure an affirmative is syllogized, in which there are not two negatives; therefore it seems that this could also happen here in the second figure. Moreover, in the first mode of the second figure an affirmative seems to be syllogized from a negative major transposed into an affirmative, thus: to whatever B belongs in no way, to all that A belongs; but C is that to which A belongs in no way; therefore B belongs to every C.
To these and similar points one must say that circular showing is made only where the conclusion is taken with the converse of one of the premises to conclude the remaining one; and in this way an affirmative cannot be syllogized in the second figure according to a circle. But if the converse of a negative is transferred into an affirmative, this will contribute nothing in this way to circular showing. In what follows, in the third mode, an affirmative will be syllogized not from the conclusion and the converse of the major transposed into an affirmative, but from the major itself so transposed, in which the whole power of inference lies, as has been said; but to syllogize in this way is not to show by a circle. And although in this way an affirmative of universal syllogisms is syllogized, as was objected, nevertheless this is not done according to a circle, as is clear from the definition of circular showing, since there the conclusion is taken with the converse of one of the premises. For if the converse of the negative transposed into an affirmative were taken, it would not serve for concluding the conclusion of the propositions circularly; rather either the very proposition translated into an affirmative is taken with the conclusion or with the converse of the conclusion. To what is objected about the negatives of the first figure, one must say that one thing is true of negatives of the first figure, and another of negatives of the second. For in the first figure the remaining premise is syllogized from the conclusion and the converse of the other premise. But in the second figure the conclusion and the converse of the negative contribute nothing to the proposed point, because an affirmative is not syllogized from the conclusion of a negative. For the precise cause why an affirmative is not syllogized here is this: circular showing is made only from the conclusion and the converse of one of the premises; but from the conclusion and the converse no syllogism is made here to the proposed point in the second figure. Thus, therefore, it is clear that circular showing is not made of an affirmative proposition in the second figure.
Privativa autem propositio circulariter in secunda figura sic ostenditur: et accipiatur primo in secundo secundae, in quo major est universalis affirmativa, minor autem universalis negativa sic: sint enim termini A B C, et medium sit A, et insit A omni B in majori, et idem A nulli insit C in minori propositione, conclusio erit in principali syllogismo, quod nulli B inest C. Si ergo sumatur conversa majoris (haec scilicet quae dicit nulli A inesse B) cum conclusione quae dicit nulli C inesse B, sequitur necessitate syllogistica nulli C inesse A: fit enim secunda figura in eodem modo, secundo scilicet secundae figurae, in quo factus est syllogismus principalis.
Si autem A B propositio major sit privativa sicut in primo modo secundae figurae, et altera propositio minor sit universalis affirmativa: tunc in circulo non observabitur eadem secunda figura, sed fit prima figura: in tali enim conjugatione in minori propositione C inerit omni A, B autem nulli inerit C, et sequitur quod B nulli inerit A, et per conversam illius ulterius sequitur, quod neque A inerit nulli B, non ergo fit hic syllogismus ex conclusione et conversa minoris ad majorem concludendam, sed et ex his et conversa majoris. Et attende quod non fit hic circulus observata eadem figura sicut in secundo modo paulo ante factum est. Et hujus ratio est, quia ex conclusione et conversa majoris secundae figurae semper fit dispositio figurae secundae: sed ex conclusione et conversa minoris semper fit dispositio figurae primae, sicut cuilibet patere potest consideranti conversas illarum propositionum, et cum conclusione sumenti eas.
Si autem non universalis sit syllogismus (sicut in tertio et quarto) illa quidem propositio quae est in toto sive universalis in tertio et quarto, non potest ostendi circulo, propter eamdem causam quae in praehabitis de syllogismis particularibus primae figurae dicta est: sed particularis syllogizatur, et praecipue in quarto, in quo universalis major est praedicativa. Sit principalis syllogismus iste, omne B est A, aliquod C non est A, ergo aliquod C non est B; sive sic, insit A omni B, C autem non omni insit A, conclusio negativa erit propositio B C, hoc est, quod aliquod C non est B: tunc ex conclusione et conversa majoris syllogizatur minor in eodem modo ejusdem figurae sic, omne A est B, aliquod C non est B, quae fuit conclusio: ergo aliquod C non est A, quae fuit minor propositio prioris syllogismi. Si ergo sumatur in conversa majoris, quod B (ut medium) insit omni A, et quod non omne C est B, sive quod non omni C inest B, sequitur alicui C A non inesse, quae fuit minor, ut diximus, in quarto secundae figurae.
Si autem major est universalis et privativa (sicut est in tertio modo secundae figurae) non ostendetur minor propositio per conclusionem et conversam majoris quae est A B, propositio universalis negativa, dicens, nullum B est A; et fiat syllogismus principalis sic, nullum B A, aliquod C est A, ergo aliquod C non est B. Patet quod hic minor propositio affirmativa est, ad quam concludendam exigitur quod utraque praemissarum sit affirmativa. Si enim cum conclusione negativa particularis sumatur conversa majoris, quae etiam negativa est, sequitur quod aut utraque praemissarum in circulo, aut altera sit negativa: et neutro modo contingit concludere particularem affirmativam qualis est minor: et idcirco non erit circularis syllogismus ad concludendam minorem. Sed haec minor aliter ostendetur per translationem negativae in affirmativam, quemadmodum in ante habitis de universalibus syllogismis dictum est in prima figura, sic scilicet quod major universalis negativa transferatur in affirmativam sic, cui alicui non inest B illi alicui inest A, sed C est cui alicui non inest B, ergo alicui C inest A, quae est minor in principali syllogismo. Sic ergo syllogismus circularis fit in secunda figura. De isto autem tertio secundae figurae hoc idem intelligendum quantum ad translationem in affirmativam, quod dictum est in quarto primae: transfertur enim in terminos particulares, eo quod particulariter concludit.
But a privative proposition is shown circularly in the second figure in this way. First let it be taken in the second mode of the second figure, in which the major is universal affirmative and the minor universal negative thus: let the terms be A B C, and let the middle be A, and let A belong to every B in the major, and the same A belong to no C in the minor proposition. The conclusion in the principal syllogism will be that C belongs to no B. Therefore, if the converse of the major is taken, namely this which says that B belongs to no A, with the conclusion which says that B belongs to no C, it follows by syllogistic necessity that A belongs to no C. For the second figure is made in the same mode, namely the second of the second figure, in which the principal syllogism was made.
But if the major proposition A B is privative, as in the first mode of the second figure, and the other proposition, the minor, is universal affirmative, then in the circle the same second figure will not be observed, but the first figure is made. For in such a conjunction, in the minor proposition C will belong to every A, but B will belong to no C, and it follows that B will belong to no A, and through its converse it further follows that neither will A belong to any B. Therefore this syllogism is not made from the conclusion and the converse of the minor for concluding the major, but both from these and from the converse of the major. And attend that here a circle is not made while the same figure is observed, as was done in the second mode a little before. The reason for this is that from the conclusion and the converse of the major of the second figure there is always made a disposition of the second figure, but from the conclusion and the converse of the minor there is always made a disposition of the first figure, as can be clear to anyone who considers the converses of those propositions and takes them with the conclusion.
But if the syllogism is not universal, as in the third and fourth modes, then that proposition which is whole or universal in the third and fourth cannot be shown by a circle, on account of the same cause which was stated in what preceded concerning particular syllogisms of the first figure. But the particular is syllogized, and especially in the fourth mode, in which the universal major is predicative. Let this be the principal syllogism: every B is A; some C is not A; therefore some C is not B. Or thus: let A belong to every B, but let C not belong to every A; the negative conclusion will be the proposition B C, that is, that some C is not B. Then from the conclusion and the converse of the major the minor is syllogized in the same mode of the same figure thus: every A is B; some C is not B, which was the conclusion; therefore some C is not A, which was the minor proposition of the prior syllogism. Therefore, if in the converse of the major it is taken that B, as middle, belongs to every A, and that not every C is B, or that B does not belong to every C, it follows that A does not belong to some C, which was the minor, as we have said, in the fourth mode of the second figure.
But if the major is universal and privative, as it is in the third mode of the second figure, the minor proposition will not be shown through the conclusion and the converse of the major, which is A B, a universal negative proposition saying no B is A. And let the principal syllogism be made thus: no B is A; some C is A; therefore some C is not B. It is clear that here the minor proposition is affirmative, and for concluding it both premises are required to be affirmative. For if the converse of the major, which is also negative, is taken with the particular negative conclusion, it follows that either both premises in the circle, or one, is negative; and in neither way is it possible to conclude the particular affirmative such as the minor is. Therefore there will not be a circular syllogism for concluding the minor. But this minor is shown otherwise, through the translation of a negative into an affirmative, as was said in what went before concerning universal syllogisms in the first figure, namely in this way: the universal negative major is transferred into an affirmative thus: to whatever particular thing B does not belong, to that particular thing A belongs; but C is that to which B does not belong; therefore A belongs to some C, which is the minor in the principal syllogism. Thus, therefore, a circular syllogism is made in the second figure. Concerning this third mode of the second figure, the same thing must be understood with respect to translation into an affirmative as was said in the fourth of the first figure; for it is transferred into particular terms, because it concludes particularly.
CAPUT IV.
Qualiter in tertia figura fit circularis ostensio.
In tertia autem figura in universalibus syllogismis (scilicet primo et secundo) quando utraeque quidem propositiones universaliter sumuntur, non contingit ostendere cum conclusione assumpta cum conversa alterius praemissarum per se invicem praemissas propositiones. Cujus causa est, quia universalis propositio non ostenditur sive concluditur, nisi per ambas praemissas universales: sed conclusio quae in circulo affirmatur cum conversa alterius est particularis, eo quod in tertia figura non concluditur nisi particulariter. Propter quod manifestum est, quoniam omnino sive penitus non contingit ostendere circulariter universalem alteram praemissarum per hanc tertiam figuram.
Si autem quis objiciat contra hoc dicens, quod in terminis convertibilibus contingit syllogizare propositionem universalem in tertia figura. Adhuc autem quia propter convertibilitatem terminorum contingit transferre particularem in universalem, sicut transfertur negativa in affirmativam: ex universali autem et conversa unius praemissarum potest concludi reliqua. Et dici solet ad hoc quod quamvis termini universalis propositionis convertibiles sint, non tamen propter hoc necesse est quod termini particulares propositionis sint convertibiles: et ideo non possunt particulares propositiones in universales transferri. Posset tamen dici, quod syllogismus circularis salvat in se formam syllogismi simpliciter: unde cum de forma syllogismi simpliciter sit, quod universalis non concluditur nisi ex universalibus: et de forma syllogismi in tertia figura sit, quod non concludat nisi particulariter: non debet syllogismus circularis in aliqua figura universalem propositionem concludere ex altera particulari: nec in tertia figura debet concludere nisi particularem, eo quod nihil debet supponi in syllogismo circulari quod sit oppositum formae syllogisticae: supponi tamen potest aliquod quod est oppositum alicui formae, cujusmodi est conversio simplex universalis affirmativae: et ideo quamvis sint termini convertibiles, non tamen debet concludi universalis in tertia figura.
Si autem aliquis dicat, quod eadem ratione potest supponi conversio particularis in universalem, sicut supponitur conversio simpliciter universalis affirmativae. Dicendum quod ad hoc quod fiat suppositio alicujus conversionis, oportet quod tantum dicatur per primam, quantum dicitur per secundam vel plus: hoc enim in omni observandum est conversione, ut tantum ponatur per convertentem, quantum ponitur per conversam vel plus. Et sic est in universali affirmativa conversa simpliciter: sed sic non esset si transferatur particularis in universalem: plus enim ponitur per universalem, quam per particularem: et ideo quamvis utraque istarum translationum transponat formam syllogismi, tamen licet alteram supponere conversionem, scilicet universalis affirmativae simpliciter, et non licet supponere alteram translationem, scilicet particularis in universalem.
CHAPTER IV.
How circular showing is made in the third figure.
But in the third figure, in universal syllogisms, namely the first and second, when both propositions are taken universally, it is not possible, with the conclusion assumed with the converse of one of the premises, to show the premise propositions through one another. The cause is that a universal proposition is not shown or concluded except through both premises universal. But the conclusion which in the circle is affirmed with the converse of the other is particular, because in the third figure nothing is concluded except particularly. On account of this it is manifest that it is absolutely or entirely impossible to show either universal premise circularly through this third figure.
But if someone objects against this, saying that in convertible terms it is possible to syllogize a universal proposition in the third figure; and again, because on account of the convertibility of the terms it is possible to transfer a particular into a universal, just as a negative is transferred into an affirmative, and from the universal and the converse of one premise the remaining one can be concluded: it is customary to say to this that, although the terms of the universal proposition are convertible, it is nevertheless not necessary on account of this that the terms of the particular proposition be convertible; and therefore particular propositions cannot be transferred into universals. Yet it could be said that the circular syllogism preserves in itself the form of a syllogism simply. Hence, since it belongs to the form of a syllogism simply that a universal is not concluded except from universals, and it belongs to the form of a syllogism in the third figure that it concludes only particularly, the circular syllogism in any figure ought not conclude a universal proposition from one particular premise; nor in the third figure ought it conclude anything except a particular, because nothing should be supposed in the circular syllogism which is opposite to syllogistic form. Nevertheless, something can be supposed which is opposite to some form, of which kind is the simple conversion of a universal affirmative. Therefore, although the terms are convertible, nevertheless a universal should not be concluded in the third figure.
But if someone says that by the same reasoning the conversion of a particular into a universal can be supposed, just as the simple conversion of a universal affirmative is supposed, one must say that in order for the supposition of some conversion to be made, it is necessary that as much be said through the first as is said through the second, or more. For this must be observed in every conversion: that as much is posited through the converting proposition as is posited through the converted one, or more. And so it is in a universal affirmative converted simply; but it would not be so if a particular is transferred into a universal, for more is posited through the universal than through the particular. Therefore, although each of these translations transposes the form of the syllogism, nevertheless it is permitted to suppose one conversion, namely the simple conversion of a universal affirmative, and it is not permitted to suppose the other translation, namely that of a particular into a universal.
Si autem quis opponat quod secundum hanc rationem non deberet supponi translatio negativae in affirmativam, quia plus ponitur per secundam quam per primam. Et dicendum quod ponitur plus per secundam quam per primam: quia negativa de se simpliciter constantiam terminorum non exigit: tamen prout est in syllogismo circulari, requirit quamdam constantiam terminorum: communicat enim cum affirmativa propositione, cujus termini constantiam habent in altero termino: idem enim dicitur per istam, nullum A est B, et per istam, cui nulli inest A illi omni inest B, secundum quod in circulo accipiuntur. Alio tamen modo dicitur per unam, et alio modo per aliam.
Si autem haec quidem propositionum universalis sit, illa vero particularis, quandoque quidem erit circularis ostensio propositionis particularis, quandoque vero non erit: quando enim minor est universalis et major particularis, tunc ostenditur circulariter particularis: quando autem major est universalis et minor particularis, tunc circulariter particularis ostendi non potest. Hoc autem ideo dictum est, quia non fit in tali ostensione circulus completus ad ostendendam particularem propositionem, nisi aliquo alio assumpto, sicut paulo post patebit. Primo tamen ostendamus hoc quod dictum est in affirmativis syllogismis. Dicamus igitur quod quando utraeque praemissae in tertia figura affirmativae sumuntur, et universalis propositio ponitur ad minorem extremitatem: tunc circularis fit ostensio particularis, quae est major propositio, sicut in tertio tertiae figurae. Quando autem universalis propositio ponitur ad majorem extremitatem, tunc non potest ostendi propositio particularis per circulum. Et ponamus primo quartum modum, in quo particularis non perfecte per circulum ostenditur, et sit primus sive principalis syllogismus. Insit enim A major extremitas omni C medio: et eidem C alicui sive particulariter insit B minor extremitas: conclusio erit quod aliquod B est A, sic, omne C est A, aliquod C est B, ergo aliquod B est A. Si ergo sumatur conversa majoris, scilicet quod omni A insit C, cum conclusione quae est, quod alicui B inest A, ostensum est per circulum quod C inest alicui B, sed conversa illius quae fuit minor prioris syllogismi (quae est quod aliquod C sit B) et illa quidem non est ostensa, sive immediate per circulum demonstrata, sic, omne A est C, aliquod B est A, ergo aliquod B est C. Hic enim est tertius primae figurae: et haec non est minor, sed conversa minoris: sed ex hac per conversionem sequitur minor, scilicet quod aliquod C est B: et quia ad illam non attingit syllogismus, ideo circulus non est complexus. Unde quod B alicui C insit, non est ostensum per circulum: quamvis necesse sit per conversionem conclusionis, si C inest alicui B, quod etiam e converso A insit alicui C. Sed tamen una conversarum non est idem omnino alii: non enim idem est hoc huic inesse, et e converso aliud isti inesse. Sed assumendum est ad circulum, quod si hoc alicui illi insit particulariter, quod etiam e converso illud isti insit: quia particularis affirmativa convertitur simpliciter. Hoc autem assumpto, tunc non fit syllogismus reflexus ex conclusione
But if someone opposes that, according to this reasoning, the translation of a negative into an affirmative ought not be supposed, because more is posited through the second than through the first, one must say that more is posited through the second than through the first, because a negative of itself simply does not require constancy of terms. Yet insofar as it is in a circular syllogism, it requires a certain constancy of terms, for it communicates with an affirmative proposition whose terms have constancy in the other term. For the same thing is said by this proposition, no A is B, and by this one, to whatever A belongs in no way, to all that B belongs, according as they are accepted in the circle. Nevertheless it is said by one in one way, and by the other in another way.
But if one of these propositions is universal and the other particular, sometimes there will indeed be a circular showing of the particular proposition, and sometimes there will not be. For when the minor is universal and the major particular, then the particular is shown circularly; but when the major is universal and the minor particular, then the particular cannot be shown circularly. This has been said because in such a showing a complete circle is not made for showing the particular proposition unless something else is assumed, as will be clear a little later. First, however, let us show what has been said in affirmative syllogisms. Therefore let us say that when both premises in the third figure are taken as affirmative, and the universal proposition is posited toward the minor extreme, then a circular showing is made of the particular proposition which is the major proposition, as in the third mode of the third figure. But when the universal proposition is posited toward the major extreme, then the particular proposition cannot be shown through a circle. Let us first posit the fourth mode, in which the particular is not perfectly shown through a circle, and let it be the first or principal syllogism. For let A, the major extreme, belong to every C, the middle; and let B, the minor extreme, belong to that same C in some instance or particularly. The conclusion will be that some B is A, thus: every C is A; some C is B; therefore some B is A. Therefore, if the converse of the major is taken, namely that C belongs to every A, with the conclusion which is that A belongs to some B, it has been shown through a circle that C belongs to some B; but this is the converse of that which was the minor of the prior syllogism, namely that some C is B, and that one indeed has not been shown, or immediately demonstrated through a circle, thus: every A is C; some B is A; therefore some B is C. For here is the third mode of the first figure, and this is not the minor but the converse of the minor; but from this, through conversion, the minor follows, namely that some C is B. And because the syllogism does not reach that proposition, the circle is not complete. Hence that B belongs to some C has not been shown through the circle, although it is necessary by conversion of the conclusion, if C belongs to some B, that conversely A also belongs to some C. Nevertheless one of the converses is not altogether the same as the other; for this belonging to that is not the same as, conversely, the other belonging to this. But for the circle it must be assumed that if this belongs particularly to that, then conversely that belongs to this, because a particular affirmative is converted simply. But when this has been assumed, then a reflexive syllogism is not made from the conclusion
Notes
- As if he were saying that it follows well: if the consequent does not exist, the antecedent does not exist; and if the antecedent does not exist, according to the adversary the consequent still exists; therefore, from first to last, upon the non-being of the consequent there follows the being of the consequent. P. J. (Quasi dicat, quod bene sequitur, si consequens non est, antecedens non est: et si antecedens non est per adversarium, adhuc consequens est: ergo a primo ad ultimum, ad non esse consequens sequitur esse consequens. P. J.) ↩
- And thus it is clear how it must be understood that the subject is related by way of a part, and the predicate by way of a whole, and how in every syllogism there is a descent from the universal to the particular. (Et sic patet quomodo intelligendum est quod subjectum se habet per modum partis, et praedicatum per modum totius, et quomodo in omni syllogismo est decursus ab universali ad particulare.) ↩
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